Problem 46 - Beginning and Intermediate Algebra

Question

Match each expression in the first column with its value in the second column. a. \((1+4) \cdot 6-3\) \(\quad\) 15 b. \(1+4 \cdot(6-3)\) \(\quad\) 13 c. \(1+4 \cdot 6-3\) \(\quad\) 27 d. \((1+4) \cdot(6-3)\) \(\quad\) 22

Step-by-Step Solution

Answer

a = 27, b = 13, c = 22, d = 15

Step 1: Evaluate Expression (1+4) * 6 - 3

First, solve the expression by evaluating inside the parenthesis \((1+4)\), which is \(5\). Now, multiply \(5\) by \(6\) to get \(30\). Finally, subtract \(3\) from \(30\) to get \(27\).

Step 2: Evaluate Expression 1 + 4 * (6 - 3)

Start by solving the expression inside the parentheses: \((6-3)\), which is \(3\). Next, multiply \(4\) by \(3\) to get \(12\) and add \(1\). The result is \(13\).

Step 3: Evaluate Expression 1 + 4 * 6 - 3

Begin by performing multiplication, \(4\times6\) which equals \(24\). Then do addition and subtraction from left to right: \(1+24-3\) equals \(22\).

Step 4: Evaluate Expression (1+4) * (6-3)

First, solve the expressions inside the parentheses: \((1+4) = 5\) and \((6-3) = 3\). Then, multiply the results \(5\) and \(3\) to get \(15\).

Step 5: Match Expressions with Values

Match each expression with its evaluated value: \((1+4) \cdot 6 - 3 = 27\), \(1 + 4 \cdot (6-3) = 13\), \(1 + 4 \cdot 6 - 3 = 22\), \((1+4) \cdot (6-3) = 15\).

Key Concepts

Parentheses

Parentheses are an essential part of mathematics as they guide you in determining which operations to perform first. When you see a mathematical expression, evaluating the terms inside parentheses is always the first step. This helps in simplifying the expression, making it more manageable.
For example, in the expression \((1+4) \cdot 6 - 3\), parentheses indicate that you should first add \(1\) and \(4\), giving \(5\). It's essential to resolve this operation before moving on to others.
Similarly, in the expression \(1 + 4 \cdot (6-3)\), you solve \((6-3)\) first, resulting in \(3\). This approach ensures accuracy and prevents errors that could occur if operations are performed in the wrong order.

• Always solve expressions inside parentheses first.
• Think of them as a 'priority box' highlighting initial operations.

Multiplication

In the order of operations, multiplication is performed after evaluating any expressions inside parentheses. It is often prioritized over addition and subtraction. This priority impacts the final result significantly.
For instance, given \(1 + 4 \cdot 6 - 3\), perform the multiplication \(4 \times 6\) first, which equals \(24\). Only after completing multiplication should you proceed with addition or subtraction which could yield different outcomes if reversed.
The expressions \((1+4) \cdot 6 - 3\) and \((1+4) \cdot (6-3)\) demonstrate how multiplication intertwines with parentheses management. First, resolve the parentheses, then multiply. This process significantly influences the result, ensuring it aligns with arithmetic rules.

• After parentheses, multiplication takes precedence.
• Ensure you carry out multiplication before moving to addition and subtraction.

Addition and Subtraction

Once any parentheses and multiplications are addressed, you focus on addition and subtraction to complete the arithmetic expression. These operations are typically executed from left to right.
Consider the expression \(1 + 4 \cdot 6 - 3\). After performing the multiplication and obtaining \(24\), continue by adding \(1\) to it, resulting in \(25\), and then subtract \(3\) to achieve the final result: \(22\).
The key to mastering addition and subtraction is keen attention to sequence. If conducted out of order, your answer may be incorrect.

• Addition and subtraction follow after multiplication.
• These operations are addressed sequentially from left to right.